arXiv subjects

Santiago Arango-Piñeros

Publications and source records attributed to Santiago Arango-Piñeros.

12 recordsLinked to original sources

arXiv2026

$7$-adic Galois representations of elliptic curves over the rationals via Kummer descent

We show that the two modular curves of level $49$ and genus $9$ left open by the work of Rouse, Sutherland and Zureick-Brown, and of Furio and Lombardo, have no non-CM rational points. Together with the theorem of Furio and Lombardo on $X_{ns}^{+}(49)$, this completes the classification of the $7$-adic images of Galois of non-CM elliptic curves over $Q$. The proof is a Kummer descent on the superelliptic equations $F(x,y) = k\,w^{7}$ of Furio and Lombardo, in the cases $7 \mid k$ that they left open. The covering curves are twists of the Fermat septic, they map to twists of the Klein quartic, and a $7$-adic computation shows that the twist attached to a solution is trivial. The rational points of the Klein quartic were determined by Hurwitz, who reduced the question to Fermat's Last Theorem for exponent $7$, proved by Lamé. Thus, the last open case of the $7$-adic part of Mazur's Program B rests on Fermat's Last Theorem for exponent $7$.

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arXiv2026

Fermat descent

Descent theory (a modern formulation of Fermat's classical method of infinite descent) is a powerful tool in arithmetic geometry. In this article, we reinterpret descent theory through the lens of quotient stacks and apply it in the setting where it first arose: the Diophantine study of generalized Fermat equations (1) \[ Ax^a + By^b + Cz^c = 0. \] We focus on understanding the arithmetic of the stacks that arise from the study of primitive integral solutions to general Fermat equations, rather than on solving any particular instance of the equation.

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arXiv2026

On $p$-adic solubility of $Ax^\ell + By^m + Cz^n = 0$

We study $p$-adic solubility of generalized Fermat equations $Ax^\ell + By^m + Cz^n = 0$ for positive integers $\ell,m,n$. For all but finitely many primes $p$, the probability of having a $p$-adic solution is described by a rational function in $p$ depending only on $\gcd(p-1,\ell,m)$, $\gcd(p-1,\ell,n)$, and $\gcd(p-1,m,n)$. When $\ell,m,n$ are pairwise coprime, we deduce that the proportion of these equations which are everywhere locally soluble is positive, given by a product of these local probabilities; when $\ell,m,n$ are not pairwise coprime, the proportion is 0\%. We then give several detailed examples demonstrating the explicit nature of the results.

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arXiv2026

Galois groups of low dimensional abelian varieties over finite fields

We consider three isogeny invariants of abelian varieties over finite fields: the Galois group, Newton polygon, and the angle rank. Motivated by work of Dupuy, Kedlaya, and Zureick-Brown, we define a new invariant called the weighted permutation representation which encompasses all three of these invariants and use it to study the subtle relationships between them. We use this permutation representation to classify the triples of invariants that occur for abelian surfaces and simple abelian threefolds.

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arXiv2026

Counting primitive integral solutions to spherical generalized Fermat equations

A solution $(x,y,z) \in \mathbb{Z}^3-\{(0,0,0)\}$ to a generalized Fermat equation \[ Ax^a + By^b + Cz^c = 0, \] is called \emph{primitive} if $\gcd(x,y,z) = 1$. By work of Beukers, we know that in the \emph{spherical} regime (that is, when the Euler characteristic $χ= \tfrac{1}{a} + \tfrac{1}{b} + \tfrac{1}{c} - 1$ is positive), if the equation has one primitive solution, then it has infinitely many. In this work, we use the method of \emph{Fermat descent}, as employed by Poonen--Schaefer--Stoll, to refine Beukers' result to an asymptotic count of the number of primitive integral solutions of bounded height.

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arXiv2026

Counting number fields of fixed degree by their smallest defining polynomial

When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of $\mathrm{GL}_2 \times \mathrm{GL}_1$ on the space of polynomials of degree $n$ so that for any two polynomials $f$ and $g$ in the same orbit, the roots of $f$ may be expressed as rational linear transformations of the roots of $g$; thus, they generate the same field. In this article, we show that almost all polynomials of degree $n$ with size at most $X$ can only define the same number field as another polynomial of degree $n$ with size at most $X$ if they lie in the same orbit for this group action. (Here we measure the size of polynomials by the greatest absolute value of their coefficients.) This improves on work of Bhargava, Shankar, and Wang, who proved a similar statement for a positive proportion of polynomials. Using this result, we prove that the number of degree $n$ fields such that the smallest polynomial defining the field has size at most $X$ is asymptotic to a constant times $X^{n+1}$ as long as $n\geq 3$. For $n = 2$, we obtain a precise asymptotic of the form $\frac{27}{π^2} X^2$.

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arXiv2025

Counting 5-isogenies of elliptic curves over $\mathbb{Q}$

We show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\cdot H^{1/6} (\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack $\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\mathbb{G}_m$-action of weights $(4, 4, 2)$.

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arXiv2025

Bounds for the relative class number problem for function fields

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the relative class number two problem in all cases where the base field has constant field not equal to $\mathbb{F}_2$.

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arXiv2025

Galois groups of simple abelian varieties over finite fields and exceptional Tate classes

We prove new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhin. Notably, our methods allow us to prove the Tate conjecture in cases when the angle rank is non-maximal. Our primary tool is a precise combinatorial condition which, given a geometrically simple abelian variety $A/\mathbf{F}_q$ with commutative endomorphism algebra, describes whether $A$ has exceptional classes (i.e., $\mathrm{Gal}( \bar{\mathbf{F}}_q/\mathbf{F}_q)$-invariant classes in $H_{\text{ét}}^{2r}(A_{\bar{\mathbf{F}}_q}, \mathbf{Q}_\ell(r))$ not contained in the span of classes of intersections of divisors). The criterion depends only on the Galois group of the minimal polynomial of Frobenius and its action on the Newton polygon of $A$. Our tools provide substantial control over the isogeny invariants of $A$, allowing us to prove a number of new results. Firstly, we provide an algorithm which, given a Newton polygon and CM field, determines if they arise from a geometrically simple abelian variety $A/\mathbf{F}_q$ and, if so, outputs one such $A$. As a consequence we show that every CM field occurs as the center of the endomorphism algebra of an abelian variety $A/\mathbf{F}_q$. Secondly, we refine a result of Tankeev and Dupuy--Kedlaya--Zureick-Brown on angle ranks of abelian varieties. In particular, we show that ordinary geometrically simple varieties of prime dimension have maximal angle rank.

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arXiv2024

Frobenius distributions of low dimensional abelian varieties over finite fields

Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_q$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre--Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre--Frobenius groups that occur for $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g>3$.

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arXiv2021

Mertens' theorem for Chebotarev sets

We generalize Mertens' product theorem to Chebotarev sets of prime ideals in Galois extensions of number fields. Using work of Rosen, we extend an argument of Williams from cyclotomic extensions to this more general case. Additionally, we compute these products for Cheboratev sets in abelian extensions, $S_3$ sextic extensions, and sets of primes represented by some quadratic forms.

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arXiv2020

Global Field Totients

Using a remainder theorem for valuations of a field, we give a new perspective on the norm function of a global field. We define the Euler totient function of a global field and recover the essential analytical properties of the classical arithmetical function, namely the product formula. In addition, we prove the holomorphicity of the associated zeta function. As an application, we recover the analog of the mean value theorem of Erdős, Dressler, and Bateman via the Weiner-Ikehara theorem.

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